Optimal. Leaf size=160 \[ \frac{(b c-a d)^2 \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{9/2}}+\frac{x (b c-a d)^3 (13 a d+3 b c)}{8 a^2 b^4 \left (a+b x^2\right )}+\frac{d^3 x (4 b c-3 a d)}{b^4}+\frac{x (b c-a d)^4}{4 a b^4 \left (a+b x^2\right )^2}+\frac{d^4 x^3}{3 b^3} \]
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Rubi [A] time = 0.197019, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {390, 1157, 385, 205} \[ \frac{(b c-a d)^2 \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{9/2}}+\frac{x (b c-a d)^3 (13 a d+3 b c)}{8 a^2 b^4 \left (a+b x^2\right )}+\frac{d^3 x (4 b c-3 a d)}{b^4}+\frac{x (b c-a d)^4}{4 a b^4 \left (a+b x^2\right )^2}+\frac{d^4 x^3}{3 b^3} \]
Antiderivative was successfully verified.
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Rule 390
Rule 1157
Rule 385
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (c+d x^2\right )^4}{\left (a+b x^2\right )^3} \, dx &=\int \left (\frac{d^3 (4 b c-3 a d)}{b^4}+\frac{d^4 x^2}{b^3}+\frac{b^4 c^4-4 a^3 b c d^3+3 a^4 d^4+4 b d (b c-a d)^2 (b c+2 a d) x^2+6 b^2 d^2 (b c-a d)^2 x^4}{b^4 \left (a+b x^2\right )^3}\right ) \, dx\\ &=\frac{d^3 (4 b c-3 a d) x}{b^4}+\frac{d^4 x^3}{3 b^3}+\frac{\int \frac{b^4 c^4-4 a^3 b c d^3+3 a^4 d^4+4 b d (b c-a d)^2 (b c+2 a d) x^2+6 b^2 d^2 (b c-a d)^2 x^4}{\left (a+b x^2\right )^3} \, dx}{b^4}\\ &=\frac{d^3 (4 b c-3 a d) x}{b^4}+\frac{d^4 x^3}{3 b^3}+\frac{(b c-a d)^4 x}{4 a b^4 \left (a+b x^2\right )^2}-\frac{\int \frac{-(b c-a d)^2 \left (3 b^2 c^2+10 a b c d+11 a^2 d^2\right )-24 a b d^2 (b c-a d)^2 x^2}{\left (a+b x^2\right )^2} \, dx}{4 a b^4}\\ &=\frac{d^3 (4 b c-3 a d) x}{b^4}+\frac{d^4 x^3}{3 b^3}+\frac{(b c-a d)^4 x}{4 a b^4 \left (a+b x^2\right )^2}+\frac{(b c-a d)^3 (3 b c+13 a d) x}{8 a^2 b^4 \left (a+b x^2\right )}+\frac{\left ((b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right )\right ) \int \frac{1}{a+b x^2} \, dx}{8 a^2 b^4}\\ &=\frac{d^3 (4 b c-3 a d) x}{b^4}+\frac{d^4 x^3}{3 b^3}+\frac{(b c-a d)^4 x}{4 a b^4 \left (a+b x^2\right )^2}+\frac{(b c-a d)^3 (3 b c+13 a d) x}{8 a^2 b^4 \left (a+b x^2\right )}+\frac{(b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.0948017, size = 160, normalized size = 1. \[ \frac{(b c-a d)^2 \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{9/2}}+\frac{x (b c-a d)^3 (13 a d+3 b c)}{8 a^2 b^4 \left (a+b x^2\right )}+\frac{d^3 x (4 b c-3 a d)}{b^4}+\frac{x (b c-a d)^4}{4 a b^4 \left (a+b x^2\right )^2}+\frac{d^4 x^3}{3 b^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 367, normalized size = 2.3 \begin{align*}{\frac{{d}^{4}{x}^{3}}{3\,{b}^{3}}}-3\,{\frac{a{d}^{4}x}{{b}^{4}}}+4\,{\frac{{d}^{3}xc}{{b}^{3}}}-{\frac{13\,{a}^{2}{x}^{3}{d}^{4}}{8\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{9\,a{x}^{3}c{d}^{3}}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{15\,{x}^{3}{c}^{2}{d}^{2}}{4\,b \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{{x}^{3}{c}^{3}d}{2\, \left ( b{x}^{2}+a \right ) ^{2}a}}+{\frac{3\,b{x}^{3}{c}^{4}}{8\, \left ( b{x}^{2}+a \right ) ^{2}{a}^{2}}}-{\frac{11\,x{a}^{3}{d}^{4}}{8\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{7\,{a}^{2}cx{d}^{3}}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{9\,a{c}^{2}x{d}^{2}}{4\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{x{c}^{3}d}{2\,b \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{5\,x{c}^{4}}{8\, \left ( b{x}^{2}+a \right ) ^{2}a}}+{\frac{35\,{a}^{2}{d}^{4}}{8\,{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{15\,ac{d}^{3}}{2\,{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{9\,{c}^{2}{d}^{2}}{4\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{c}^{3}d}{2\,ab}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,{c}^{4}}{8\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.7646, size = 1689, normalized size = 10.56 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.71684, size = 513, normalized size = 3.21 \begin{align*} - \frac{\sqrt{- \frac{1}{a^{5} b^{9}}} \left (a d - b c\right )^{2} \left (35 a^{2} d^{2} + 10 a b c d + 3 b^{2} c^{2}\right ) \log{\left (- \frac{a^{3} b^{4} \sqrt{- \frac{1}{a^{5} b^{9}}} \left (a d - b c\right )^{2} \left (35 a^{2} d^{2} + 10 a b c d + 3 b^{2} c^{2}\right )}{35 a^{4} d^{4} - 60 a^{3} b c d^{3} + 18 a^{2} b^{2} c^{2} d^{2} + 4 a b^{3} c^{3} d + 3 b^{4} c^{4}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{a^{5} b^{9}}} \left (a d - b c\right )^{2} \left (35 a^{2} d^{2} + 10 a b c d + 3 b^{2} c^{2}\right ) \log{\left (\frac{a^{3} b^{4} \sqrt{- \frac{1}{a^{5} b^{9}}} \left (a d - b c\right )^{2} \left (35 a^{2} d^{2} + 10 a b c d + 3 b^{2} c^{2}\right )}{35 a^{4} d^{4} - 60 a^{3} b c d^{3} + 18 a^{2} b^{2} c^{2} d^{2} + 4 a b^{3} c^{3} d + 3 b^{4} c^{4}} + x \right )}}{16} - \frac{x^{3} \left (13 a^{4} b d^{4} - 36 a^{3} b^{2} c d^{3} + 30 a^{2} b^{3} c^{2} d^{2} - 4 a b^{4} c^{3} d - 3 b^{5} c^{4}\right ) + x \left (11 a^{5} d^{4} - 28 a^{4} b c d^{3} + 18 a^{3} b^{2} c^{2} d^{2} + 4 a^{2} b^{3} c^{3} d - 5 a b^{4} c^{4}\right )}{8 a^{4} b^{4} + 16 a^{3} b^{5} x^{2} + 8 a^{2} b^{6} x^{4}} + \frac{d^{4} x^{3}}{3 b^{3}} - \frac{x \left (3 a d^{4} - 4 b c d^{3}\right )}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10662, size = 343, normalized size = 2.14 \begin{align*} \frac{{\left (3 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 60 \, a^{3} b c d^{3} + 35 \, a^{4} d^{4}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{2} b^{4}} + \frac{3 \, b^{5} c^{4} x^{3} + 4 \, a b^{4} c^{3} d x^{3} - 30 \, a^{2} b^{3} c^{2} d^{2} x^{3} + 36 \, a^{3} b^{2} c d^{3} x^{3} - 13 \, a^{4} b d^{4} x^{3} + 5 \, a b^{4} c^{4} x - 4 \, a^{2} b^{3} c^{3} d x - 18 \, a^{3} b^{2} c^{2} d^{2} x + 28 \, a^{4} b c d^{3} x - 11 \, a^{5} d^{4} x}{8 \,{\left (b x^{2} + a\right )}^{2} a^{2} b^{4}} + \frac{b^{6} d^{4} x^{3} + 12 \, b^{6} c d^{3} x - 9 \, a b^{5} d^{4} x}{3 \, b^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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